In Cut-model inference, feedback from the suspect module is completely cut. However, there may be a bias-variance trade-off: if the parameters of a well-specified module are poorly informed by “local" information then limited information from misspecified modules may allow us to bring the uncertainty down without introducing significant bias.
Semi-Modular Inference (\(\eta \) -SMI, [1]}) generalises Cut-model inference as it offers a means by which we can control the influence of suspect modules on the fit for a good module. Candidate posterior distributions are indexed by an associated influence parameter \(\eta \) . At \(\eta =1\)  \(\eta \) -SMI is standard Bayesian inference and at \(\eta =0\)  \(\eta \) -SMI reproduces the Cut-model. [1]} suggest choosing \(\eta \)  maximizing the expected log pointwise predictive density (ELPD) though this choice is not an essential part of their method and other criteria [3]} may be more appropriate in different settings. [4]} adapt \(\eta \) -SMI for Geographically Weighted Regression using an influence parameter across likelihood factors which is modeled as a function of distance between the spatial observation locations.
